In offices, shops and other workplaces the employer is the responsible person even where the building is leased. In residential blocks the role lands on whoever controls the common parts: the freeholder, a resident-owned management company, a right to manage company or a managing agent, depending on how the building is run. Mixed-use buildings routinely have several responsible persons at once, such as the shop employer on the ground floor and the freeholder above. Where a building changes hands or managers, the duty follows whoever has control at the time, which is why handovers should transfer the assessment and its action plan explicitly.
The responsible person rarely writes the assessment personally. They appoint a competent assessor, and who can do a fire risk assessment for flats covers what that competence looks like for a block. The appointment does not move the duty: as who is legally responsible for a risk assessment explains, accountability stays with the role holder, and the record must now name whoever did the work.